Thermodynamic derivation of the fractional Fokker–Planck equation for fractal turbulent chaos with power memory

A stochastic-thermodynamic approach to the derivation of the generalized fractional Fokker–Planck–Kolmogorov (FPK) equations is considered. The equations describe turbulent transfer processes in a subsystem of turbulent chaos on the basis of fractional dynamics, which takes into account the structure and metric of fractal time. The actual turbulent motion of a fluid is known to be intermittent, since it demonstrates the properties that are intermediate between the properties of regular and chaotic motions. On the other hand, the process of the flow turbulization may be non-Markovian because of the multidimensional spatiotemporal correlations of pulsating parameters; in a physical language, this means that the process has a memory. The introduction of fractional time derivatives into the FPK kinetic equations, used to find the probability distribution functions for different statistical characteristics of structured turbulence, makes it possible to use an unified mathematical formalism in considering the effects of memory, non-locality, and time intermittence, with which we usually associate the presence of turbulent bursts against the background of less intense low-frequency oscillations in the background turbulence.